From Graphs to Hypergraphs: Hypergraph Projection and its Remediation

TL;DR

This work addresses the loss of higher-order information when replacing hypergraphs with their graph projections. It provides a theoretical analysis that identifies two common loss-inducing patterns and proves that exact recovery from projections is generically impossible without extra information. To remediate, the authors propose SHyRe, a learning-based framework that leverages a domain-specific training hypergraph to reconstruct higher-order structures from projections, using a ρ(n,k)-alignment statistic, a budgeted clique sampler, and a feature-rich hyperedge classifier. Empirically, SHyRe substantially outperforms baselines across eight real-world datasets, and enables improved downstream tasks such as protein ranking, link prediction, and clustering, illustrating the practical value of reconstructed hypergraphs as informative intermediate representations.

Abstract

We study the implications of the modeling choice to use a graph, instead of a hypergraph, to represent real-world interconnected systems whose constituent relationships are of higher order by nature. Such a modeling choice typically involves an underlying projection process that maps the original hypergraph onto a graph, and is common in graph-based analysis. While hypergraph projection can potentially lead to loss of higher-order relations, there exists very limited studies on the consequences of doing so, as well as its remediation. This work fills this gap by doing two things: (1) we develop analysis based on graph and set theory, showing two ubiquitous patterns of hyperedges that are root to structural information loss in all hypergraph projections; we also quantify the combinatorial impossibility of recovering the lost higher-order structures if no extra help is provided; (2) we still seek to recover the lost higher-order structures in hypergraph projection, and in light of (1)'s findings we propose to relax the problem into a learning-based setting. Under this setting, we develop a learning-based hypergraph reconstruction method based on an important statistic of hyperedge distributions that we find. Our reconstruction method is evaluated on 8 real-world datasets under different settings, and exhibits consistently good performance. We also demonstrate benefits of the reconstructed hypergraphs via use cases of protein rankings and link predictions.

Figures (21)

• Figure 1: hypergraph reconstruction as the reversal of hypergraph projection. Given projected graph $G_1$, the goal is to reconstruct the original hypergraph $\mathcal{H}_1$ in real world as accurately as possible.
• Figure 2: The upper panel shows hyperedge patterns that trigger errors in max-clique based reconstruction. Error I leads to missing $E_2$ (false negative), while Error II results in an incorrect identification of max clique ${v_1, v_3, v_5}$ as a hyperedge (false positive) and overlooks ${v_1, v_5}$ (false negative). The lower panel shows the errors' associations with $\mathcal{E}$ and $\mathcal{M}$.
• Figure 3: $\mathcal{E}$ is the set of hyperedges; $\mathcal{E}'$ is the set of hyperedges not nested in any other hyperedges; $\mathcal{M}$ is the set of maximal cliques in $G$. Error I, II result from the violation of Conditions I, II, respectively. Error I $=\frac{|\mathcal{E}\backslash\mathcal{E}'|}{|\mathcal{E}\cup\mathcal{M}|}$, Error II $=\frac{|\mathcal{M}\backslash\mathcal{E}'|+ |\mathcal{E}'\backslash\mathcal{M}|}{|\mathcal{E}\cup\mathcal{M}|}$. Errors caused by violation of both conditions are counted as Error I.
• Figure 4: (a) Supervised hypergraph reconstruction. $\mathcal{H}_0$ and $\mathcal{H}_1$ belong to the same application domain. Given $\mathcal{H}_0$ (and its projection $G_0$), the task is to reconstruct $\mathcal{H}_1$ from its projection $G_1$. (b) 4-step reconstruction: (1) the clique sampler is optimized on $G_0$ and $\mathcal{H}_0$; (2) the clique sampler samples candidates from $G_0$ and $G_1$, then passes result to the hyperedge classifier; (3) the hyperedge classifier extracts features of candidates from $G_0$ and trains on them; (4) the hyperedge classifier extracts features of candidates from $G_1$ and identify hyperedges.
• Figure 5: (a) $\rho(n, k)$-alignment on dataset Enron; $\mathcal{H}_0$, $\mathcal{H}_1$ obtained by splitting all emails by a middle timestamp. (b) $\rho(n, k)$-alignment on more datasets. Notice the column-wise similarity and row-wise difference.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Definition 1
• Theorem 3
• Theorem 4
• proof
• proof
• Lemma 5
• proof
• Lemma 6